The cherry blossom is out in force in a small street on Paris's southern outskirts. Well it might be, because it is here in an old shoe factory that the political sensation of France's presidential election has been stitched together.

The headquarters of Jean-Luc Mélenchon's campaign is known as The Factory. One visit is enough to convince you that there is more to Mélenchon's rise in the polls than the man himself. He is difficult to typify. He talks to people who think they belong to the modern world, but about subjects they are not used to hearing in an election like this – history, France, culture – why they inherited the welfare state they have.

The former Socialist senator is not just a firebrand on the stump. He is also everyone's favourite professeur de fac, a real instituteur républicain. But Mélenchon, the candidate who speaks his mind, is not enough to explain the rise of the Front de Gauche, a coalition of seven parties of which by far the strongest are the communists. Mélenchon's own party, the Parti de Gauche is dwarfed by it (10,000 members compared with the communist PCF's 130,000).

Irrespective of how it does on Sunday in the first round of the presidential election, it harbours serious long-term political ambitions. It already thinks it has won one national election – the referendum on the doomed European constitution. This was in 2005 and the year in which Mélenchon broke Socialist party ranks, sacrificing a comfortable career as a senator over this issue. Although what was later to become known as the Lisbon treaty was little more than a compendium of previous treaties, in the left's eyes it institutionalised a concept of Europe that put free-trade principles guaranteeing an internal market above a Europe defending social and trade union rights.

Mélenchon campaigned for a no vote, and got it. Brussels did not accept no for an answer and when the constitution was downsized into a more humble treaty, it was to the French parliament, not to the popular vote, that it turned for approval. For the French far left this still counted as a victory, because they had recaptured this form of Euroscepticism from the far right.

The next stop on this political journey was Germany and the founding congress of Die Linke, which was formed out of defections from the SPD and the dissolution of the former communists. Raquel Garrido, in charge of the front's international relations, said: "The European social democratic movement had an historic opportunity in 1989 to rebuild something that was both democratic and socialist. And they failed. So what we had on our hands was a double responsibility – to build something after the failure of state communism and the European social democratic movement."

It took them two more years to organise their split from the French Socialists and get the communists on board. From then on they have risen steadily in national prominence, first at the European parliamentary elections in 2009 when they got five seats in Strasbourg, and then the French regional elections. When they started their presidential campaign on June 29 last year, they decided to go big – large open-air rallies that had more the feel of Tahrir Square than a sedate and controlled indoor rally designed for television. Their slogan, "Place au Peuple", was a pun in that it means both "make room for the people" and the "the people's square", but after Mélenchon appeared on national television, that is literally what his rallies became. More than 100,000 people crammed into Place de la Bastille in Paris, and similar scenes were repeated in Nantes and Toulouse.

After these shows of force, both Nicolas Sarkozy and François Hollande felt they had to organise mass open-air rallies themselves. But the Front de Gauche was not really targeting them. They decided early on to go for Marine Le Pen, Jean-Marie's daughter who had transformed the Front National into a modern far-right party.

It has been a deadly battle. After a television duel in which Mélenchon destroyed Le Pen's credentials as a feminist, particularly on the issue of abortion, the Front National went back to its far right roots, plucking one of their heros out of the air. He is Robert Brasillach, a fascist who advocated that Jewish children should be sent to the camps with their families. He was the only journalist to be executed after the libération at the end of the second world war, but as he did not kill anyone, he is honoured to this day by Le Pen as a victim of conscience.

The Front National's fortunes have wavered, and recently they have been going up in the polls into third place. But the Front de Gauche is unwavering in its intention to recapture the working-class vote. Garrido said: "We really need the extreme right to go back into their box. This would be one of the real transformations of this campaign. If we go in front of the Front National, France would go back to what it really is, a very mixed country with a high rate of mixed marriages, its own republican culture and tradition." Whatever happens on Sunday, the Front de Gauche is here to stay.